# Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers

arXiv: https://arxiv.org/abs/2503.04684 · HTML: https://mlsanity.com/paper/2503.04684/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 6 Mar 2025.

## Builds on (most cited first, 14 of 14)

- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Calibrated Adaptive Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.08202/) (2020)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [Convergence Rates of Gaussian ODE Filters](https://mlsanity.com/paper/1807.09737/) (2018)
- [Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systems](https://mlsanity.com/paper/2002.09301/) (2020)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems](https://mlsanity.com/paper/2008.08737/) (2020)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models](https://mlsanity.com/paper/2211.11103/) (2022)

## Abstract (arXiv)

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.